The lengths of the tangent drawn from any point on the circle $15x^{2} + 15y^{2} - 48x + 64y = 0$ to the two circles $5x^{2} + 5y^{2} - 24x + 32y + 75 = 0$ and $5x^{2} + 5y^{2} - 48x + 64y + 300 = 0$ are in the ratio of
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$1 : 2$
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$2 : 3$
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$3 : 4$
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$None\ of\ these$
A
Correct answer
Explanation
The power of a point (x, y) with respect to a circle x^2+y^2+2gx+2fy+c=0 is x^2+y^2+2gx+2fy+c. The circle 15x^2+15y^2-48x+64y=0 is x^2+y^2-3.2x+4.26y=0. The two circles are C1: x^2+y^2-4.8x+6.4y+15=0 and C2: x^2+y^2-9.6x+12.8y+60=0. The ratio of the powers of any point on the first circle to C1 and C2 is constant. Calculating the ratio of the constants or radii shows the ratio of lengths is 1:2.