Finding the Value of k for a Circle Tangent Length
The length of the tangent from an external point $(x_1, y_1)$ to a circle defined by the equation $x^2 + y^2 + 2gx + 2fy + c = 0$ is given by the formula:
Length = $\sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$
Alternatively, if the circle equation is given in the form $x^2 + y^2 + Dx + Ey + F = 0$, the length is $\sqrt{x_1^2 + y_1^2 + Dx_1 + Ey_1 + F}$.
Applying the Tangent Length Formula
We are given:
- Point $(x_1, y_1) = (2, 5)$
- Circle equation: $x^2 + y^2 - 5x + 4y + k = 0$. Here, $D = -5$, $E = 4$, and $F = k$.
- Length of the tangent = $\sqrt{37}$
Substitute the values into the formula:
$\sqrt{37} = \sqrt{(2)^2 + (5)^2 - 5(2) + 4(5) + k}$
Calculating the Value of k
To find $k$, we first square both sides of the equation:
$37 = (2)^2 + (5)^2 - 5(2) + 4(5) + k$
Evaluate the terms:
$37 = 4 + 25 - 10 + 20 + k$
Combine the constant terms on the right side:
$37 = (4 + 25 + 20) - 10 + k$
$37 = 49 - 10 + k$
$37 = 39 + k$
Isolate $k$:
$k = 37 - 39$
$k = -2$
Therefore, the value of $k$ is -2.
Similar Questions
Study the given pattern carefully and select the number from among the given options that can replace the question mark (?).
Study the given pattern carefully and select the numbers from among the given options that would be most appropriate for replacing the question mark (?)
- Study the given pattern carefully and select the number from among the given options that can replace the question mark (?).
$2$ $9$ $5$ $3$ $8$ ? $4$ $7$ $6$ $24$ $504$ $30$ Study the given pattern carefully and select the number that can replace the question mark (?) in it.
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
Study the given pattern carefully and select the number from among the given options that can replace the question mark (?).
Study the given pattern carefully and select the number from the given options that can replace the question mark (?).
Study the given pattern carefully and select the number from among the given options that can replace the question mark (?).
Study the given pattern carefully and select the number that can replace the question mark (?) in it
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
Important Questions from Missing Number
Study the given pattern carefully and select the number from among the given options that can replace the question mark (?) in it.
13 16 52 15 24 90 23 36 ? Select the set in which the numbers are related in the same way as are the numbers of the given set.
(NOTE: Operations should be performed on the whole numbers, without breaking down the number into its constituent digits.E.g. 13 - Operations on 13 such as adding /Subtracting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
(25, 150, 3)
(18, 180, 5)
Select the option that is related to the fifth number in the same way as the second number is related to the first number and the fourth number is related to the third number.
7 ∶ 35 ∶∶ 21 ∶ 399 ∶∶ 4 ∶ ?
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
First row - 12, 7, 32
Second row - 17, 13, 33
Third row - 14, 8, ?
(NOTE : Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 - Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is NOT allowed)
Select the number that can replace the question mark (?) in the following pattern.
5
6
7
8
9
115
206
?
502
719
