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 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 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 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 the third figure.
Study the given pattern carefully and select the number that can replace the question mark (?) in the third figure.
Study the given pattern carefully and select the number that can replace the question mark (?) in the third figure.
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
Important Questions from Missing Number
Find the missing number from the below options.
Find the missing number from the below options.
Find the missing number from the below options.
In the following question, select the number which can be placed at the sign of question mark (?) from the given alternatives
12
17
121
21
81
144
36
45
?
In the following question, find the missing number from the given series.
4, 12, 48, 240, 1440, ?
