Find the equation to the pair of straight lines joining the origin to the intersections oi the straight line `y=mx + c` and the curve `x^2 + y^2=a^2` . Prove that they are at right angles if `2c^2=a^2(1+m^2)`.

Explanation

The combined equation of OP and OQ is obtained by homogenising the equation `x^(2)+y^(2)=a^(2)` by6 means of the equation `y=mx+c.`
`therefore (y-mx)/(c)=1`
Hence the combined equation of OP and OQ is
`x^(2)+y^(2)=a^(2)((y-mx)/(c))^(2)`
`i.e., (1-(m^(2)a^(2))/(c^(2)))x^(2)+(2ma^(2))/(c^(2))xy+(1-(a^(2))/(c^(2))y^(2))=0`
The two straight lines are at right angles if
`1-(m^(2)a^(2))/(c^(2))+1-(a^(2))/(c^(2))=0`
`i.e., 2=(a^(2))/(c^(2))(1+m^(2))`
`i.e., 2c^(2)=a^(2)(1+m^(2))`